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eventual property

Let XXX be a set and PPP a property on the elements of XXX. Let (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} be a net (DDD a directed set) in XXX (that is, xi∈XsubscriptxiXx_{i}\in X). As each xi∈XsubscriptxiXx_{i}\in X, xisubscriptxix_{i} either has or does not have property PPP. We say that the net (xi)subscriptxi(x_{i}) has property PPPabovej∈DjDj\in D if xisubscriptxix_{i} has property PPP for all i≥jiji\geq j. Furthermore, we say that (xi)subscriptxi(x_{i})eventually has property PPP if it has property PPP above some j∈DjDj\in D.

Examples.

1.

Let AAA and BBB be non-empty sets. For x∈AxAx\in A, let P⁢(x)PxP(x) be the property that x∈BxBx\in B. So PPP is nothing more than the property of elements being in the intersection of AAA and BBB. A net (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} eventually has PPPmeans that for some j∈DjDj\in D, the set {xi∣i∈A⁢, ⁢i≥j}⊆Bconditional-setsubscriptxiiA, ijB\{x_{i}\mid i\in A\mbox{, }i\geq j\}\subseteq B. If D=ℤDℤD=\mathbb{Z}, then we have that AAA and BBBeventually coincide.

2.

Now, suppose AAA is a topological space, and BBB is an openneighborhood of a pointx∈AxAx\in A. For y∈AyAy\in A, let PB⁢(y)subscriptPByP_{B}(y) be the property that y∈ByBy\in B. Then a net (xi)subscriptxi(x_{i}) has PBsubscriptPBP_{B} eventually for every neighborhood BBB of xxx is a characterization of convergence (to the point xxx, and xxx is the accumulation point of (xi)subscriptxi(x_{i})).

3.

If AAA is a poset and B={x}⊆ABxAB=\{x\}\subseteq A. For y∈AyAy\in A, let P⁢(y)PyP(y) again be the property that y=xyxy=x. Let (xi)subscriptxi(x_{i}) be a net that eventually has property PPP. In other words, (xi)subscriptxi(x_{i}) is eventually constant. In particular, if for every chainDDD, the net (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} is eventually constant in AAA, then we have a characterization of the ascending chain condition in AAA.

4.

directed net. Let RRR be a preorder and let (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} be a net in RRR. Let x⁢(D)xDx(D) be the image of the net: x⁢(D)={xi∈R∣i∈D}xDconditional-setsubscriptxiRiDx(D)=\{x_{i}\in R\mid i\in D\}. Given a fixedk∈DkDk\in D and some y∈x⁢(D)yxDy\in x(D), let Pk⁢(y)subscriptPkyP_{k}(y) be the property (on x⁢(D)xDx(D)) that xk≤ysubscriptxkyx_{k}\leq y. Let

S={k∈D∣(xi)⁢ eventually has ⁢Pk}.Sconditional-setkDsubscriptxi eventually has subscriptPkS=\{k\in D\mid(x_{i})\mbox{ eventually has }P_{k}\}.

If S=DSDS=D, then we say that the net (xi)subscriptxi(x_{i}) is directed, or that (xi)subscriptxi(x_{i}) is a directed net. In other words, a directed net is a net (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} such that for everyi∈DiDi\in D, there is a k⁢(i)∈DkiDk(i)\in D, such that xi≤xjsubscriptxisubscriptxjx_{i}\leq x_{j} for all j≥k⁢(i)jkij\geq k(i).

If (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} is a directed net, then x⁢(D)xDx(D) is a directed set: Pick xi,xj∈x⁢(D)subscriptxisubscriptxjxDx_{i},x_{j}\in x(D), then there are k⁢(i),k⁢(j)∈DkikjDk(i),k(j)\in D such that xi≤xmsubscriptxisubscriptxmx_{i}\leq x_{m} for all m≥k⁢(i)mkim\geq k(i) and xj≤xnsubscriptxjsubscriptxnx_{j}\leq x_{n} for all n≥k⁢(j)nkjn\geq k(j). Since DDD is directed, there is a t∈DtDt\in D such that t≥k⁢(i)tkit\geq k(i) and t≥k⁢(j)tkjt\geq k(j). So xt≥xk⁢(i)≥xisubscriptxtsubscriptxkisubscriptxix_{t}\geq x_{{k(i)}}\geq x_{i} and xt≥xk⁢(j)≥xjsubscriptxtsubscriptxkjsubscriptxjx_{t}\geq x_{{k(j)}}\geq x_{j}.

However, if (xi)i∈DsubscriptsubscriptxiiD(x_{i})_{{i\in D}} is a net such that x⁢(D)xDx(D) is directed, (xi)subscriptxi(x_{i}) need not be a directed net. For example, let D={p,q,r}DpqrD=\{p,q,r\} such that p≤q≤rpqrp\leq q\leq r, and R={a,b}RabR=\{a,b\} such that a≤baba\leq b. Define a net x:D→Rnormal-:xnormal-→DRx:D\to R by x⁢(p)=x⁢(r)=bxpxrbx(p)=x(r)=b and x⁢(q)=axqax(q)=a. Then xxx is not a directed net.

Remark. The eventual property is a property on the class of nets (on a given set XXX and a given property PPP). We can write Eventually⁡(P,X)EventuallyPX\operatorname{Eventually}(P,X) to denote its dependence on XXX and PPP.